Integral representation of Euler - Mascheroni Constant

[Yuqing]

The definition of the Euler - Mascheroni constant, [itex]\gamma[/itex], is given as
[tex]\gamma = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k} - \ln(n)[/tex]
or equivalently in integral form as [tex]\gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx[/tex]

I saw a seeming related integral representation
[tex]\gamma = 1 - \int_{1}^{\infty} \frac{x - \left\lfloor x\right\rfloor}{x^2}\ dx[/tex]
but I can't seem to derive it. I was wondering if anyone can shed some light on this.

 

[Citan Uzuki]

The easiest way to do this is to just expand out the integral, like so:

[tex]\begin{align}&1 - \int_{1}^{\infty} \frac{x-\lfloor x \rfloor}{x^2}\ dx \\ &= 1 - \lim_{n \rightarrow \infty} \int_{1}^{n} \frac{x-\lfloor x \rfloor}{x^2}\ dx \\ &= \lim_{n \rightarrow \infty} 1 - \sum_{k=1}^{n-1} \int_{k}^{k+1} \frac{x - k}{x^2}\ dx \\ &= \lim_{n \rightarrow \infty} 1 - \sum_{k=1}^{n-1} \left( \ln x + \frac{k}{x}\right) \Big|_{k}^{k+1} \\ &=\lim_{n \rightarrow \infty} 1 - \sum_{k=1}^{n-1} \left( \ln (k+1) - \ln k + \frac{k}{k+1} - 1\right) \\ &=\lim_{n \rightarrow \infty} 1 - \sum_{k=1}^{n-1} \left( \ln (k+1) - \ln k - \frac{1}{k+1}\right) \\ &=\lim_{n \rightarrow \infty} 1 - \sum_{k=1}^{n-1} (\ln (k+1) - \ln k) + \sum_{k=1}^{n-1} \frac{1}{k+1} \\ &= \lim_{n \rightarrow \infty} 1 - \ln n + \sum_{k=2}^{n} \frac{1}{k} \\ &=\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k} - \ln n = \gamma \end{align}[/tex]

 

[Yuqing]

Hmm, that's quite a novel approach. Thank you.

 

[Dickfore]

I think this integral is related to the Euler-Mascheroni Constant as well:

[tex]
\int_{0}^{\infty}{\ln{t} \, e^{-t} \, dt} = -\gamma
[/tex]

 

[CellCoree]

I find the integral representation of the Euler-Mascheroni constant to be a fascinating mathematical concept. It shows how this important constant, which arises in various mathematical and scientific fields, can be expressed in terms of an infinite sum or integral.

The first integral representation given, \gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx, is derived from the definition of the Euler-Mascheroni constant as the limit of a partial sum. This representation highlights the connection between the constant and the harmonic numbers, which are defined as the sum of the reciprocals of the first n positive integers. As n increases, the difference between the harmonic number and the natural logarithm of n approaches the Euler-Mascheroni constant.

The second integral representation, \gamma = 1 - \int_{1}^{\infty} \frac{x - \left\lfloor x\right\rfloor}{x^2}\ dx, is another way to express the Euler-Mascheroni constant in terms of an integral. This representation is derived by considering the integral of the fractional part of x, which is defined as x - \left\lfloor x\right\rfloor. This integral evaluates to 1, and when subtracted from 1, we get the Euler-Mascheroni constant.

While I cannot provide a full derivation of this second representation, I can offer some insights into its connection to the first representation. Both representations involve the fractional part of x, but in the first representation, it is in the denominator, while in the second representation, it is in the numerator. This difference in placement leads to the difference in the integral expressions.

Overall, these integral representations of the Euler-Mascheroni constant demonstrate the versatility and complexity of this important mathematical constant. They also highlight the interconnectedness of different mathematical concepts and how they can be used to express and understand fundamental constants in our universe.